On the word problem for just infinite groups

A group is just infinite if it is infinite and every proper quotient {of it} is finite. In this talk, we consider the word problem for just infinite groups given by recursively enumerable sets of relations. For the finitely generated case we show that the word problem is uniformly decidable. In the countably generated case, the situation is more delicate, as decidability may depend on the chosen presentation. We prove that the word problem is (non-uniformly) decidable in most cases, with the exception of locally finite groups analogous to the algorithmically finite groups of Miasnikov and Osin. For these exceptions, we construct specific presentations where the word problem is undecidable, alongside standard presentations of the same groups for which the word problem remains decidable.